Expression, p6

Average Error: 3.7 → 2.6

Time: 32.3s

Precision: 64

\[-14 \le a \le -13 \land -3 \le b \le -2 \land 3 \le c \le 3.5 \land 12.5 \le d \le 13.5\]

Math

\[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2\]

↓

\[2 \cdot \sqrt[3]{\log \left(e^{\left(b + \left(a + d\right)\right) + c}\right) \cdot \left(\left(\left(\left(b + c\right) + d\right) + a\right) \cdot \left(\left(\left(b + c\right) + d\right) + a\right)\right)}\]

C

double f(double a, double b, double c, double d) {
        double r12382636 = a;
        double r12382637 = b;
        double r12382638 = c;
        double r12382639 = d;
        double r12382640 = r12382638 + r12382639;
        double r12382641 = r12382637 + r12382640;
        double r12382642 = r12382636 + r12382641;
        double r12382643 = 2.0;
        double r12382644 = r12382642 * r12382643;
        return r12382644;
}

↓

double f(double a, double b, double c, double d) {
        double r12382645 = 2.0;
        double r12382646 = b;
        double r12382647 = a;
        double r12382648 = d;
        double r12382649 = r12382647 + r12382648;
        double r12382650 = r12382646 + r12382649;
        double r12382651 = c;
        double r12382652 = r12382650 + r12382651;
        double r12382653 = exp(r12382652);
        double r12382654 = log(r12382653);
        double r12382655 = r12382646 + r12382651;
        double r12382656 = r12382655 + r12382648;
        double r12382657 = r12382656 + r12382647;
        double r12382658 = r12382657 * r12382657;
        double r12382659 = r12382654 * r12382658;
        double r12382660 = cbrt(r12382659);
        double r12382661 = r12382645 * r12382660;
        return r12382661;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	3.7
Target	3.8
Herbie	2.6

\[\left(a + b\right) \cdot 2 + \left(c + d\right) \cdot 2\]

Derivation

1. Initial program 3.7

   \[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2\]
2. Using strategy rm

3. Applied associate-+r+ 2.8

   \[\leadsto \left(a + \color{blue}{\left(\left(b + c\right) + d\right)}\right) \cdot 2\]
4. Using strategy rm

5. Applied add-cbrt-cube 2.9

   \[\leadsto \color{blue}{\sqrt[3]{\left(\left(a + \left(\left(b + c\right) + d\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)}} \cdot 2\]
6. Using strategy rm

7. Applied add-log-exp 2.9

   \[\leadsto \sqrt[3]{\left(\left(a + \left(\left(b + c\right) + d\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)\right) \cdot \left(a + \left(\left(b + c\right) + \color{blue}{\log \left(e^{d}\right)}\right)\right)} \cdot 2\]

8. Applied add-log-exp 2.9

   \[\leadsto \sqrt[3]{\left(\left(a + \left(\left(b + c\right) + d\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)\right) \cdot \left(a + \left(\left(b + \color{blue}{\log \left(e^{c}\right)}\right) + \log \left(e^{d}\right)\right)\right)} \cdot 2\]

9. Applied add-log-exp 2.9

   \[\leadsto \sqrt[3]{\left(\left(a + \left(\left(b + c\right) + d\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)\right) \cdot \left(a + \left(\left(\color{blue}{\log \left(e^{b}\right)} + \log \left(e^{c}\right)\right) + \log \left(e^{d}\right)\right)\right)} \cdot 2\]

10. Applied sum-log 2.8

    \[\leadsto \sqrt[3]{\left(\left(a + \left(\left(b + c\right) + d\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)\right) \cdot \left(a + \left(\color{blue}{\log \left(e^{b} \cdot e^{c}\right)} + \log \left(e^{d}\right)\right)\right)} \cdot 2\]

11. Applied sum-log 2.7

    \[\leadsto \sqrt[3]{\left(\left(a + \left(\left(b + c\right) + d\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)\right) \cdot \left(a + \color{blue}{\log \left(\left(e^{b} \cdot e^{c}\right) \cdot e^{d}\right)}\right)} \cdot 2\]

12. Applied add-log-exp 2.7

    \[\leadsto \sqrt[3]{\left(\left(a + \left(\left(b + c\right) + d\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)\right) \cdot \left(\color{blue}{\log \left(e^{a}\right)} + \log \left(\left(e^{b} \cdot e^{c}\right) \cdot e^{d}\right)\right)} \cdot 2\]

13. Applied sum-log 2.6

    \[\leadsto \sqrt[3]{\left(\left(a + \left(\left(b + c\right) + d\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)\right) \cdot \color{blue}{\log \left(e^{a} \cdot \left(\left(e^{b} \cdot e^{c}\right) \cdot e^{d}\right)\right)}} \cdot 2\]

14. Simplified 2.9

    \[\leadsto \sqrt[3]{\left(\left(a + \left(\left(b + c\right) + d\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)\right) \cdot \log \color{blue}{\left(e^{\left(d + \left(a + b\right)\right) + c}\right)}} \cdot 2\]
15. Using strategy rm

16. Applied associate-+r+ 2.6

    \[\leadsto \sqrt[3]{\left(\left(a + \left(\left(b + c\right) + d\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)\right) \cdot \log \left(e^{\color{blue}{\left(\left(d + a\right) + b\right)} + c}\right)} \cdot 2\]

17. Final simplification 2.6

    \[\leadsto 2 \cdot \sqrt[3]{\log \left(e^{\left(b + \left(a + d\right)\right) + c}\right) \cdot \left(\left(\left(\left(b + c\right) + d\right) + a\right) \cdot \left(\left(\left(b + c\right) + d\right) + a\right)\right)}\]

Reproduce

herbie shell --seed 2019124 
(FPCore (a b c d)
  :name "Expression, p6"
  :pre (and (<= -14 a -13) (<= -3 b -2) (<= 3 c 3.5) (<= 12.5 d 13.5))

  :herbie-target
  (+ (* (+ a b) 2) (* (+ c d) 2))

  (* (+ a (+ b (+ c d))) 2))